Do two circles always have a radical axis?

Question

Do two circles always have a radical axis?

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I came across this question in my book.I think every pair of circles have a radical axis,but my book answer says NO,not every pair of circles have a radical axis.
Can someone please elaborate with example which two circles do not have a radical axis associated with them.Thanks.

Answer 1

Concentric circles do not have a associated radical axis. One way to think about why this is the case stems from the fact that the configuration has radial symmetry, but a radical axis is a line, hence it does not have radial symmetry. Hence a radical axis does not exist.

If two circles in general position intersect, the radical axis is the line joining the points of intersection.

Here is a proof for concentric circles of different sizes.

Let $P$ be a point on the radical axis outside the circles. It follows that the tangents from $P$ to the circles would be the same length.

Let $O$ be the center of these circles. $OP^2=r_1^2+l^2=r_2^2+l^2$, where $r_1, r_2$ are the radius of the circle and $l$ is the length of the tangent.

As we assumed $r_1\neq r_2$, it follows that there is no solution, i.e. there is no radical axis which lies outside the circles.

Answer 2

The equation of radical axis of a pair of circles is given by $S-S'=0$.

For concentric circles we will get $S-S'= constant$ instead of line equations. Hence we can conclude that all pair of circles wont necessarily have a radical axis.